Given the sequence # { 6,18,54,162,... } #, find the next three terms and an expression for the #n^(th)# term?

1 Answer
Feb 21, 2017

Answer:

The relationship is that the #n^{th)# term is given by (assuming we start the sequence at #n=1#:

# u_n = 6xx3^(n-1) #

the next three terms are:

# 486, 1458 ,4374#

Explanation:

The sequence:

# { 6,18,54,162,... } #

clearly is not linear or quadratic as the terms increase too rapidly, we note that they are all even and factors of #6#, so we let us see how factoring out #6# helps to establish the pattern:

# { 6,6xx3,6xx9,6xx27,... } #

and we can now see that the factor of #6# is multiplied by a power of #3#, giving us:

# { 6,6xx3^1,6xx3^2,6xx3^3,... } #

And we also now that #3^0 = 1#, so we can modify the first term as follows:

# { 6xx3^0,6xx3^1,6xx3^2,6xx3^3,... } #

And then we can see that the pattern is established and that the #n^{th)# term is given by (assuming we start the sequence at #n=1#:

# u_n = 6xx3^(n-1) #

So let us check that this works, and then form the next three terms:

# n=1 => u_1=6xx3^0 = 6 #
# n=2 => u_2=6xx3^1 = 18 #
# n=3 => u_3=6xx3^2 = 54 #
# n=4 => u_4=6xx3^3 = 162 #

# n=5 => u_5=6xx3^4 = 486#
# n=6 => u_6=6xx3^5 = 1458#
# n=7 => u_7=6xx3^6 = 4374#